Quasi-particle interferometry for logical gates

ABSTRACT

A quantum computer can only function stably if it can execute gates with extreme accuracy. “Topological protection” is a road to such accuracies. Quasi-particle interferometry is a tool for constructing topologically protected gates. Assuming the corrections of the Moore-Read Model for ν=5/2&#39;s FQHE (Nucl. Phys. B 360, 362 (1991)) we show how to manipulate the collective state of two e/4-charge anti-dots in order to switch said collective state from one carrying trivial SU(2) charge, |1&gt;, to one carrying a fermionic SU(2) charge |ε&gt;. This is a NOT gate on the {|1&gt;, |ε&gt;} qubit and is effected by braiding of an electrically charged quasi particle σ which carries an additional SU(2)-charge. Read-out is accomplished by σ-particle interferometry.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a division of U.S. patent application Ser. No.11/544,492, filed on Oct. 6, 2006, which is a division of U.S. patentapplication Ser. No. 11/233,653, filed on Sep. 23, 2005, now U.S. Pat.No. 7,250,624. The disclosure of each of the above-referenced patentapplications is incorporated herein by reference.

GOVERNMENT RIGHTS

This invention was supported in part by funds from the U.S. Government(National Science Foundation Grant No. DMR-0411800 and Army ResearchOffice Grant No. W911NF-04-1-236). The U.S. Government, therefore, mayhave certain rights in this invention.

BACKGROUND OF THE INVENTION

Since the discovery of the fractional quantum Hall effect (FQHE) in1982, topological phases of electrons have been a subject of greatinterest. Many abelian topological phases have been discovered in thecontext of the quantum Hall regime. More recently, high-temperaturesuperconductivity and other complex materials have provided the impetusfor further theoretical studies of and experimental searches for abeliantopological phases. The types of microscopic models admitting suchphases are now better understood. Much less is known about non-abeliantopological phases. They are reputed to be obscure and complicated, andthere has been little experimental motivation to consider non-abeliantopological phases. However, non-abelian topological states would be anattractive milieu for quantum computation.

It has become increasingly clear that if a new generation of computerscould be built to exploit quantum mechanical superpositions, enormoustechnological implications would follow. In particular, solid statephysics, chemistry, and medicine would have a powerful new tool, andcryptography also would be revolutionized.

The standard approach to quantum computation is predicated on thequantum bit (“qubit”) model in which one anticipates computing on alocal degree of freedom such as a nuclear spin. In a qubit computer,each bit of information is typically encoded in the state of a singleparticle, such as an electron or photon. This makes the informationvulnerable. If a disturbance in the environment changes the state of theparticle, the information is lost forever. This is known asdecoherence—the loss of the quantum character of the state (i.e., thetendency of the system to become classical). All schemes for controllingdecoherence must reach a very demanding and possibly unrealizableaccuracy threshold to function.

Topology has been suggested to stabilize quantum information. Atopological quantum computer would encode information not in theconventional zeros and ones, but in the configurations of differentbraids, which are similar to knots but consist of several differentthreads intertwined around each other. The computer would physicallyweave braids in space-time, and then nature would take over, carryingout complex calculations very quickly. By encoding information in braidsinstead of single particles, a topological quantum computer does notrequire the strenuous isolation of the qubit model and represents a newapproach to the problem of decoherence.

In 1997, there were independent proposals by Kitaev and Freedman thatquantum computing might be accomplished if the “physical Hilbert space”V of a sufficiently rich TQFT (topological quantum field theory) couldbe manufactured and manipulated. Hilbert space describes the degrees offreedom in a system. The mathematical construct V would need to berealized as a new and remarkable state for matter and then manipulatedat will.

The computational power of a quantum mechanical Hilbert space ispotentially far greater than that of any classical device. However, itis difficult to harness it because much of the quantum informationcontained in a system is encoded in phase relations which one mightexpect to be easily destroyed by its interactions with the outside world(“decoherence” or “error”). Therefore, error-correction is particularlyimportant for quantum computation. Fortunately, it is possible torepresent information redundantly so that errors can be diagnosed andcorrected.

An interesting analogy with topology suggests itself: local geometry isa redundant way of encoding topology. Slightly denting or stretching asurface such as a torus does not change its genus, and small puncturescan be easily repaired to keep the topology unchanged. Only largechanges in the local geometry change the topology of the surface.Remarkably, there are states of matter for which this is more than justan analogy. A system with many microscopic degrees of freedom can haveground states whose degeneracy is determined by the topology of thesystem. The excitations of such a system have exotic braidingstatistics, which is a topological effective interaction between them.Such a system is said to be in a topological phase. The unusualcharacteristics of quasiparticles in such states can lead to remarkablephysical properties, such as a fractional quantized Hall conductance.Such states also have intrinsic fault-tolerance. Since the ground statesare sensitive only to the topology of the system, interactions with theenvironment, which are presumably local, cannot cause transitionsbetween ground states unless the environment supplies enough energy tocreate excitations which can migrate across the system and affect itstopology. When the temperature is low compared to the energy gap of thesystem, such events will be exponentially rare.

A different problem now arises: if the quantum information is sowell-protected from the outside world, then how can we—presumably partof the outside world—manipulate it to perform a computation? The answeris that we must manipulate the topology of the system. In this regard,an important distinction must be made between different types oftopological phases. In the case of those states which are Abelian, wecan only alter the phase of the state by braiding quasiparticles. In thenon-Abelian case, however, there will be a set of g>1 degenerate states,ψ_(a), a=1, 2, . . . , g of particles at x₁, x₂, . . . , x_(n).Exchanging particles 1 and 2 might do more than just change the phase ofthe wave function. It might rotate it into a different one in the spacespanned by the ψ_(a)S:ψ_(a)→M_(ab) ¹²ψ_(b)   (1)

On the other hand, exchanging particles 2 and 3 leads to ψ_(a)→M_(ab)²³ψ_(b). If M_(ab) ¹² and M_(ab) ²³ do not commute (for at least somepairs of particles), then the particles obey non-Abelian braidingstatistics. In the case of a large class of states, the repeatedapplication of raiding transformations M_(ab) ^(ij) allows one toapproximate any desired unitary transformation to arbitrary accuracyand, in this sense, they are universal quantum computers. Unfortunately,no non-Abelian topological states have been unambiguously identified sofar. Some proposals have been put forward for how such states mightarise in highly frustrated magnets, where such states might bestabilized by very large energy gaps on the order of magnetic exchangecouplings, but the best prospects in the short run are in quantum Hallsystems, where Abelian topological phases are already known to exist.

The best candidate is the quantized Hall plateau with

$\sigma_{ab} = {\frac{5}{2}{\frac{{\mathbb{e}}^{2}}{h}.}}$The 5/2 fractional quantum Hall state (as well as its particle-holesymmetric analog, the 7/2 state) is now routinely observed inhigh-quality (i.e., low-disorder) samples. In addition, extensivenumerical work using finite-size diagonalization and wavefunctionoverlap calculations indicates that the 5/2 state belongs to thenon-Abelian topological phase characterized by a Pfaffian quantum Hallwavefunction. The set of transformations generated by braidingquasiparticle excitations in the Pfaffian state is not computationallyuniversal (i.e., is not dense in the unitary group), but othernon-Abelian states in the same family are. Thus, it is important to (a)determine if the ν=5/2 state is, indeed, in the Pfaffian universalityclass and, if so, to (b) use it to store and manipulate quantuminformation.

SUMMARY OF THE INVENTION

An experimental device as described herein can address both of theaforementioned determinations. Features of such a device are inspired byanti-dot experiments measuring the charge of quasiparticles in Abelianfractional quantum Hall states such as ν=1/3 and proposals for measuringtheir statistics. Our measurement procedure relies upon quantuminterference as in the electronic Mach-Zehnder interferometer in whichAharonov-Bohm oscillations were observed in a two dimensional electrongas.

In order to establish which topological phase the ν=5/2 plateau is in,one must directly measure quasiparticle braiding statistics. Remarkably,this has never been done even in the case of the usual ν=1/3 quantumHall plateau (although in this case, unlike in the ν=5/2 case,computational solutions of small systems leave little doubt about whichtopological phase the plateau is in). Part of the problem is that it isdifficult to disentangle the phase associated with braiding from thephase which charged particles accumulate in a magnetic field.Ironically, it may actually be easier to measure the effect ofnon-Abelian braiding statistics because it is not just a phase and istherefore qualitatively different from the effect of the magnetic field.

A logical gate according to the invention enables the manipulation of acollective quantum state of one or more anti-dots disposed in afractional quantum Hall effect (FQHE) fluid. A FQHE fluid is an exoticform of matter that arises when electrons at the flat interface of twosemiconductors are subjected to a powerful magnetic field and cooled totemperatures close to absolute zero. The electrons on the flat surfaceform a disorganized liquid sea of electrons, and if some extra electronsare added, quasi-particles called anyons emerge. Quasi-particles areexcitations of electrons, and, unlike electrons or protons, anyons canhave a charge that is a fraction of a whole number.

Anti-dots are “holes” in the FQHE fluid created by charge, i.e., islandsof higher potential where the FQHE fluid does not exist. Anti-dots arenot quasi particles, per se, but can have a quasi-particle charge. Thecollective quantum state of the one or more anti-dots may be a statecarrying trivial SU(2) charge |1>, or a state carrying a fermionic SU(2)charge |ε>.

The collective state of the quasi-particles may be read out by drawingan output current out of the quantum Hall fluid. The value of the outputcurrent will indicate whether the collective state is |1> or |ε>. Thestate can be the state of a single anti-dot, or the collective state oftwo or more anti-dots.

To operate the logical gate, i.e., to flip between states |1> and |ε>, aσ quasi-particle may be caused to tunnel by adjusting the electricalpotential on conductive gates that are provided for adjusting electricalpotentials confining a fractional quantum Hall fluid. Tunneling of thequasi-particle may deform the contours of the FQHE fluid. It should beunderstood that quasi-particles tunnel more easily than electronsbecause quasi-particles have less charge being in essence electronfractions.

As described in detail below, quasi-particle interferometry byelectrically charged particles may be used to build logical gates andread out for quantum computers, as realized in a system for implementinga NOT-gate in ν=5/2 FQHE systems.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1D depict links formed by taking a quasiparticle around a qubitpair.

FIG. 2 depicts an example embodiment of a NOT gate for a quantumcomputer.

FIG. 3 is a block diagram showing an example computing environment inwhich aspects of the invention may be implemented.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The Pfaffian state may be viewed as a quantum Hall state of p—wavepaired fermions. The quasiparticles in this phase have charge −e/4 (note/2, as one might naively assume from the Landau-level filling fraction

${v = {2 + \frac{1}{2}}};$this emphasizes the importance of an experiment, such as described in V.J. Goldman and B. Su, Science 267, 1010 (1995), to measure thequasiparticle charge at ν=5/2) When there are 2n quasiparticles at fixedpositions in the system, there is a 2^(n−1)-dimensional degenerate spaceof states. Exchanging and braiding quasiparticles is related to theaction of the 2n-dimensional Clifford algebra on this space, as hasrecently been confirmed by direct numerical evaluation of the Berrymatrices. In particular, two charge −e/4 quasiparticles can “fuse” toform a charge −e/2 quasiparticle either with or without a neutralfermion in its core. One may view the charge −e/2 quasiparticle as thequantum Hall incarnation of a superconducting vortex with a fermioniczero mode in its core. We will regard the presence or absence of aneutral fermion in this core state if the two charge −e/4 quasiparticleswere fused as our qubit. So long as the two quasiparticles are kept farapart, the neutral fermion is not localized anywhere and, therefore, thequbit is unmeasurable by any local probe or environment. However, we canmeasure the qubit by encircling it with a charge −e/4 quasiparticle. Thepresence of the neutral fermion causes the state to acquire an extrafactor of −1 during this process. The qubit can also be manipulated bytaking another charge −e/4 quasiparticle between the two charge −e/4quasiparticles comprising the qubit, i.e., around one but not the other.Such a process transforms a state without a neutral fermion into a statewith one and vice versa. Thus, it flips the qubit (and also multipliesby i). By performing an experiment which measures this qubit, flips it,and then re-measures it, we can demonstrate that the ν=5/2 state is in anon-Abelian topological phase. A few additional similar experiments maybe necessary to fully nail down that it is in the Pfaffian phase ratherthan another non-Abelian phase. Such an experiment can only work if theenvironment does not flip the qubit before we have a chance to measureit, so the success of this experiment would demonstrate the stability ofa topological qubit in a non-Abelian quantum Hall state. By varying thetime between measurements, one could determine the decoherence time ofthe qubit in order to quantitatively compare it with other approaches toquantum computation.

The claimed quasiparticle braiding properties can be seen from the formof the four-quasihole wavefunctions given in C. Nayak and F. Wilczek,Nucl. Phys. B 479, 529 (1996). The ground state wavefunction takes theform

$\begin{matrix}{{\Psi_{g.s.}\left( z_{j} \right)} = {\prod\limits_{j < k}{\left( {z_{j} - z_{k}} \right)^{2}{\prod\limits_{j}{{\mathbb{e}}^{{- {z_{j}}^{2}}/4} \cdot {{{Pf}\left( \frac{1}{z_{j} - z_{k}} \right)}.}}}}}} & (2)\end{matrix}$where the Pfaffian is the square root of the determinant of anantisymmetric matrix. If we write

$\begin{matrix}{{\Psi_{{(13)}{(24)}}\left( z_{j} \right)} = {\prod\limits_{j < k}{\left( {z_{j} - z_{k}} \right)^{2}{\prod\limits_{j}{{\mathbb{e}}^{{- {z_{j}}^{2}}/4} \times {{Pf}\left( \frac{{\left( {z_{j} - \eta_{1}} \right)\left( {z_{j} - \eta_{3}} \right)\left( {z_{k} - \eta_{2}} \right)\left( {z_{k} - \eta_{4}} \right)} + \left( j\leftrightarrow k \right)}{z_{j} - z_{k}} \right)}}}}}} & (3)\end{matrix}$and similarly for Ψ₍₁₄₎₍₂₃₎, then the four-quasihole wavefunctions canbe written in a basis in which their braiding is completely explicit:

$\begin{matrix}{{\Psi^{({0,1})}\left( z_{j} \right)} = {\frac{\left( {\eta_{13}\eta_{24}} \right)^{\frac{1}{4}}}{\left( {1 \pm \sqrt{x}} \right)^{\frac{1}{2}}}\left( {\Psi_{{(13)}{(24)}} \pm {\sqrt{x}\Psi_{{(14)}{(23)}}}} \right)}} & (4)\end{matrix}$where η₁₃=η₁−η₃, etc. and x=η₁₄η₂₃/η₁₃η₂₄. Let us suppose that thequasiholes at η₁ and η₂ form our qubit. The quasiholes at η₃ and η₄ willbe used to measure and manipulate them. From (4), we see that taking η₃around η₁ and η₂ results in a factor i in the state Ψ⁽⁰⁾ but −i in thestate Ψ⁽¹⁾. Taking η₃ around either η₁ or η₂ (but not both) transformsΨ⁽⁰⁾ into iΨ⁽¹⁾ and vice versa.

As shown in FIGS. 1A-1D, it is also possible to verify the logicassociated to braiding operations using a few formal properties of theJones polynomial at q=exp(πi/4). Taking one quasiparticle around thequbit pair (i.e., “linking”) results in an extra −1 if the qubit is instate |1

(a factor d=−q−q⁻¹ also arises regardless of whether or not thequasiparticle encircles the qubit). The boxed “1” is a projector on thepair of quasiparticles that puts them in the state |1>. By evaluatingthe Jones polynomial at q=exp(πi/4) for these links, desired matrixelements for braiding operations manipulating the qubit can be obtained.The Jones polynomial (operator) at q=exp(πi/4) vanishes for the links inFIGS. 1A and 1B by calculation, and for the links in FIG. 1C by parity.The Jones polynomial at q=exp(πi/4) is non-vanishing only for the linksin FIG. 1D, which applies to all processes with topologically-equivalentlink diagrams, e.g., interchanging inputs/outputs so, for example, FIG.1D corresponds to four different processes. In the links depicted inFIG. 1D, the qubit is flipped by the elementary braid operation.

FIG. 2 depicts a plan view of an example embodiment of a NOT gateaccording to the invention. The basic setup which is proposed is aquantum Hall bar with two individually-gated anti-dots in its interior,labeled 1 and 2 in FIG. 2. The plan view shown in FIG. 2 may be about3-5 μm×about 3-5 μm. A fractional quantum Hall effect (FQHE) fluid(which may include some 10 million electrons, for example, may be formedin a gallium arsenide hetero-junction, i.e., at a crystal interfacebetween a gallium arsenide crystal, for example, and an aluminum galliumarsenide crystal.

Electrically-conductive gates enable tunneling between the two edges atM, N and P,Q, thereby allowing a measurement of the qubit formed by thecorrelation between anti-dots 1 and 2. Electrically-conductive gatesalso allow tunneling at A, B that flips the qubit. An electricalpotential may be applied to enable tunneling of quasi-particles betweenA and B. It may be useful to have a third anti-dot 3 disposed between Aand B in order to precisely control the charge which tunnels between Aand B. Because there is only a probability that a tunneling event willoccur, the anti-dot between A and B may be used to indicate whether atunneling event has actually occurred. One can determine whether atunneling event has occurred by using coulomb blockade, for example, orsome other experimental method to “watch” an e/4 charge first jump onand then jump off the anti-dot disposed between gates A and B. It shouldbe understood that the gates may be front gates (i.e., gates disposed ontop of the crystal) or back gates (gates disposed on the bottom of thecrystal). If back gates are used, the potential would be the opposite ofthat used with front gates.

There are three basic procedures which we would like to execute: (1)initialize the qubit and measure its initial state, (2) flip the qubit,and (3) measure it again. In order to initialize the qubit, we first putcharge e/2 on one of the anti-dots, say 1. Since the fermionic zero modeis now localized on this anti-dot, the environment will “measure” it,and it will either be occupied or unoccupied (not a superposition of thetwo). We can determine which state it is in by applying voltage to thefront gates at M and N and at P and Q so that tunneling can occur therewith amplitudes t_(MN) and t_(PQ). The longitudinal conductivity, σ_(xx)is determined by the probability for current entering the bottom edge atX in FIG. 2 to exit along the top edge at Y. This is given, to lowestorder in t_(MN) and t_(PQ), by the interference between two processes:one in which a quasiparticle tunnels from M to N, and another in whichthe quasiparticle instead continues along the bottom edge to P, tunnelsto Q, and then moves along the top edge to N. We subsume into t_(PQ) thephase associated with the extra distance traveled in the second process.The relative phase of these processes depends on the state of the qubit.If a neutral fermion is not present, which we will denote by |0

, then σ_(xx) ∝|t_(MN)+i t_(PQ)|². If it is present, however,which wedenote by |1

, then σ_(xx) ∝|t_(MN)−i t_(PQ)|². We take the visibility ofAharonov-Bohm oscillations in a device with similar limitations (e.g.,the possibility of the tunneling quasiparticles becoming dephased bytheir interaction with localized two-level systems) as an indicationthat our proposed read-out procedure will work.

Without loss of generality, let us suppose that the initial state of thequbit is |0

. Now, let us apply voltage to anti-dots 1 and 2 so that charge e/4 istransferred from 1 to 2. There is now one charge −e/4 quasihole on eachanti-dot. The state of the qubit is unaffected by this process. In orderto flip this qubit, we now apply voltage to the front gates at A and Bso that one charge e/4 quasiparticle tunnels between the edges. In orderto ensure that only a single quasiparticle tunnels, it is useful to tunethe voltage of the anti-dot at C and the backgate at A so that a singlequasiparticle tunnels from the edge to the anti-dot at C. If theanti-dot is small, its charging energy will be too high to allow morethan one quasiparticle to tunnel at once. We can then lower the voltageof the backgate at A so that no further tunneling can occur there andapply voltage to the backgate at B so that the quasiparticle can tunnelfrom C to B. By this two-step process, we can tunnel a singlequasiparticle from A to B. If the ν=5/2 plateau is in the phase of thePfaffian state, this will transform |0

to |1

. This is our logical NOT operation. The gate which creates the anti-dotat C must be turned off at the beginning and end of the bit flip processso that there are no quasiparticles there either before or after whichcould become entangled with our qubit.

We can now measure our qubit again by tuning the front gates so thattunneling again occurs between M and N and between P and Q withamplitudes t_(MN) and t_(PQ). If, as expected, the qubit is now in thestate |1

we will find σ_(xx)α|t_(MN)−i t_(PQ)|². On the other hand, if the ν=5/2state were Abelian, σ_(xx) would not be affected by the motion of aquasiparticle from A to B.

In order to execute these steps, it is important that we know that wehave one (modulo 4) charge −e/4 quasihole on each anti-dot. This can beensured by measuring the tunneling conductance G_(t) ^(ad) from one edgeto the other through each anti-dot. As we sweep the magnetic field,there will be a series of peaks in G_(t) ^(ad) corresponding to thepassage through the Fermi level of quasihole states of the anti-dot. Thespacing ΔB between states is determined by the condition that anadditional state passes through the Fermi level when one additionalhalf-flux-quantum, Φ₀/2 is enclosed in the dot. Thus, the number ofquasiholes is given simply by └B/ΔB┘. Alternatively, with a back gate,we could directly measure capacitatively the charge on each anti-dot. Ifthe back gate voltage is V_(BG) (relative to the zero quasihole casewhen the gate defining the anti-dot is turned off), then the charge onthe anti-dot is q=εV_(BG)A/d, where A=Φ₀/2ΔB is the area of the dot, othe dielectric constant, and d the distance between the back gate andthe 2DEG.

Estimate of Error Rate. Bit flip and phase flip errors, respectively,occur when an uncontrolled charge −e/4 quasiparticle performs one of thetwo basic processes above: encircling one of the anti-dots (or passingfrom one edge to the other between them) or encircling both of them. Therate for these processes is related to the longitudinal resistivity(which vanishes within experimental accuracy) because it is limited bythe density and mobility of excited quasiparticles. Even withoutconsidering the suppression factor associated with the latter (whichdepends on the ratio of the diffusion or hopping length, a, to thesystem size, L), we already have a strong upper bound on the error ratefollowing from the thermally activated form of the former (in k_(B)=1units):

$\begin{matrix}{\frac{\Gamma}{\Delta} \sim {\frac{T}{\Delta}{\mathbb{e}}^{{- \Delta}/T}} < 10^{- 30}} & (5)\end{matrix}$

Here, we have used the best current measured value for the quasiparticlegap Δ=500 mk of the 5/2 state and the lowest achieved measurementtemperature T=5 mK. For arbitrary braid-based computation, in a moreelaborate device, it is sufficient if we further have e^(Δ/T)>νΔ L²,where ν is the density-of-states. The effect of residual pinnedquasiparticles can be diagnosed and accounted for in software. Theseerror rates are substantially lower than the estimated error rate forany other physical implementations of quantum computation in anyproposed architectures. Compared to other scalable solid statearchitectures, such as localized electron spin qubits in Si or GaAsnanostructures, where the estimated error rate is around 10⁻⁴ even inthe best possible circumstances, the errors associated with ν=5/2quantum Hall anyons is essentially negligible. This miniscule error ratearises from the intrinsic robustness of the topological phase which isfundamentally immune to all local environmental perturbations.

The ideal error rate for the 5/2 state may actually be substantiallylower than even this very low currently achievable value of 10⁻³⁰. Thereis strong theoretical evidence that the ideal excitation gap (˜2K) forthe 5/2 quantum Hall state is much larger than the currently achievedgap value of 500 mK. Using an ideal gap of 2K, we get an astronomicallylow error rate of 10⁻¹⁰⁰. This expected higher value of Δ (˜2K) isconsistent with the experimental development of the activation gapmeasurement of the 5/2 state. The early measurements on fairly modestquality samples (i.e., relatively highly disordered) gave Δ˜100 mK,whereas recent measurements in extremely high-quality (i.e., lowdisorder) samples give Δ˜300-500 mK. This implies that the 5/2excitation gap is susceptible to strong suppression by disorder as hasrecently been theoretically argued. Since improvement in sample qualityhas already led to a factor of 5 enhancement in Δ (from 100 mK to 500mK), it is not unreasonable to expect further improvements.

There are, in principle, other sources of error, but we expect them tobe of minor significance. For example, if two quasiparticles come closeto each other, then their mutual interaction leads to an error (e.g.,through the exchange of a virtual particle). Such a virtual exchange is,however, a quantum tunneling process which should be exponentiallysuppressed. Therefore, keeping the quasiparticles reasonably far fromeach other should essentially eliminate this error.

We note that, although we have discussed only the 5/2 Pfaffian quantizedHall state throughout this paper, our considerations and arguments applyequally well to the experimentally often-observed 7/2 quantized Hallstate which, being the “hole” analog of the 5/2 state by virtue of theparticle-hole symmetry, should have equivalent topological andnon-Abelian properties. We believe the 5/2 state to be a betterexperimental candidate for topological quantum computation because themeasured excitation gap in the 5/2 state tends to be much higher thanthat in the 7/2 state. We should also mention that recently the 12/5fractional quantum Hall state has been observed experimentally in thehighest mobility sample at the lowest possible temperatures. This state,thought to be a non-Abelian state related to parafermions, isparticularly exciting from the perspective of topological quantumcomputation because its braid group representation is dense in theunitary group making this state an ideal candidate for topologicalquantum computation. The measured gap value in the 12/5 state iscurrently rather small (˜70 mK). We expect that this is also stronglyaffected by disorder and that the eventual ideal gap at 12/5 will bemuch larger.

Example Computing Environment

FIG. 3 and the following discussion are intended to provide a briefgeneral description of a suitable computing environment in which anexample embodiment of the invention may be implemented. It should beunderstood, however, that handheld, portable, and other computingdevices of all kinds are contemplated for use in connection with thepresent invention. While a general purpose computer is described below,this is but one example. The present invention also may be operable on athin client having network server interoperability and interaction.Thus, an example embodiment of the invention may be implemented in anenvironment of networked hosted services in which very little or minimalclient resources are implicated, e.g., a networked environment in whichthe client device serves merely as a browser or interface to the WorldWide Web.

Although not required, the invention can be implemented via anapplication programming interface (API), for use by a developer ortester, and/or included within the network browsing software which willbe described in the general context of computer-executable instructions,such as program modules, being executed by one or more computers (e.g.,client workstations, servers, or other devices). Generally, programmodules include routines, programs, objects, components, data structuresand the like that perform particular tasks or implement particularabstract data types. Typically, the functionality of the program modulesmay be combined or distributed as desired in various embodiments.Moreover, those skilled in the art will appreciate that the inventionmay be practiced with other computer system configurations. Other wellknown computing systems, environments, and/or configurations that may besuitable for use with the invention include, but are not limited to,personal computers (PCs), automated teller machines, server computers,hand-held or laptop devices, multi-processor systems,microprocessor-based systems, programmable consumer electronics, networkPCs, minicomputers, mainframe computers, and the like. An embodiment ofthe invention may also be practiced in distributed computingenvironments where tasks are performed by remote processing devices thatare linked through a communications network or other data transmissionmedium. In a distributed computing environment, program modules may belocated in both local and remote computer storage media including memorystorage devices.

FIG. 3 thus illustrates an example of a suitable computing systemenvironment 100 in which the invention may be implemented, although asmade clear above, the computing system environment 100 is only oneexample of a suitable computing environment and is not intended tosuggest any limitation as to the scope of use or functionality of theinvention. Neither should the computing environment 100 be interpretedas having any dependency or requirement relating to any one orcombination of components illustrated in the exemplary operatingenvironment 100.

With reference to FIG. 3, an example system for implementing theinvention includes a general purpose computing device in the form of acomputer 110. Components of computer 110 may include, but are notlimited to, a processing unit 120, a system memory 130, and a system bus121 that couples various system components including the system memoryto the processing unit 120. The system bus 121 may be any of severaltypes of bus structures including a memory bus or memory controller, aperipheral bus, and a local bus using any of a variety of busarchitectures. By way of example, and not limitation, such architecturesinclude Industry Standard Architecture (ISA) bus, Micro ChannelArchitecture (MCA) bus, Enhanced ISA (EISA) bus, Video ElectronicsStandards Association (VESA) local bus, and Peripheral ComponentInterconnect (PCI) bus (also known as Mezzanine bus).

Computer 110 typically includes a variety of computer readable media.Computer readable media can be any available media that can be accessedby computer 110 and includes both volatile and nonvolatile, removableand non-removable media. By way of example, and not limitation, computerreadable media may comprise computer storage media and communicationmedia. Computer storage media includes both volatile and nonvolatile,removable and non-removable media implemented in any method ortechnology for storage of information such as computer readableinstructions, data structures, program modules or other data. Computerstorage media includes, but is not limited to, random access memory(RAM), read-only memory (ROM), Electrically-Erasable ProgrammableRead-Only Memory (EEPROM), flash memory or other memory technology,compact disc read-only memory (CDROM), digital versatile disks (DVD) orother optical disk storage, magnetic cassettes, magnetic tape, magneticdisk storage or other magnetic storage devices, or any other mediumwhich can be used to store the desired information and which can beaccessed by computer 110. Communication media typically embodiescomputer readable instructions, data structures, program modules orother data in a modulated data signal such as a carrier wave or othertransport mechanism and includes any information delivery media. Theterm “modulated data signal” means a signal that has one or more of itscharacteristics set or changed in such a manner as to encode informationin the signal. By way of example, and not limitation, communicationmedia includes wired media such as a wired network or direct-wiredconnection, and wireless media such as acoustic, radio frequency (RF),infrared, and other wireless media. Combinations of any of the aboveshould also be included within the scope of computer readable media.

The system memory 130 includes computer storage media in the form ofvolatile and/or nonvolatile memory such as ROM 131 and RAM 132. A basicinput/output system 133 (BIOS), containing the basic routines that helpto transfer information between elements within computer 110, such asduring start-up, is typically stored in ROM 131. RAM 132 typicallycontains data and/or program modules that are immediately accessible toand/or presently being operated on by processing unit 120. By way ofexample, and not limitation, FIG. 3 illustrates operating system 134,application programs 135, other program modules 136, and program data137. RAM 132 may contain other data and/or program modules.

The computer 110 may also include other removable/non-removable,volatile/nonvolatile computer storage media. By way of example only,FIG. 3 illustrates a hard disk drive 141 that reads from or writes tonon-removable, nonvolatile magnetic media, a magnetic disk drive 151that reads from or writes to a removable, nonvolatile magnetic disk 152,and an optical disk drive 155 that reads from or writes to a removable,nonvolatile optical disk 156, such as a CD ROM or other optical media.Other removable/non-removable, volatile/nonvolatile computer storagemedia that can be used in the example operating environment include, butare not limited to, magnetic tape cassettes, flash memory cards, digitalversatile disks, digital video tape, solid state RAM, solid state ROM,and the like. The hard disk drive 141 is typically connected to thesystem bus 121 through a non-removable memory interface such asinterface 140, and magnetic disk drive 151 and optical disk drive 155are typically connected to the system bus 121 by a removable memoryinterface, such as interface 150.

The drives and their associated computer storage media discussed aboveand illustrated in FIG. 3 provide storage of computer readableinstructions, data structures, program modules and other data for thecomputer 110. In FIG. 3, for example, hard disk drive 141 is illustratedas storing operating system 144, application programs 145, other programmodules 146, and program data 147. Note that these components can eitherbe the same as or different from operating system 134, applicationprograms 135, other program modules 136, and program data 137. Operatingsystem 144, application programs 145, other program modules 146, andprogram data 147 are given different numbers here to illustrate that, ata minimum, they are different copies. A user may enter commands andinformation into the computer 110 through input devices such as akeyboard 162 and pointing device 161, commonly referred to as a mouse,trackball or touch pad. Other input devices (not shown) may include amicrophone, joystick, game pad, satellite dish, scanner, or the like.These and other input devices are often connected to the processing unit120 a-f through a user input interface 160 that is coupled to the systembus 121, but may be connected by other interface and bus structures,such as a parallel port, game port or a universal serial bus (USB).

A monitor 191 or other type of display device is also connected to thesystem bus 121 via an interface, such as a video interface 190. Inaddition to monitor 191, computers may also include other peripheraloutput devices such as speakers 197 and printer 196, which may beconnected through an output peripheral interface 195.

The computer 110 may operate in a networked environment using logicalconnections to one or more remote computers, such as a remote computer180. The remote computer 180 may be a personal computer, a server, arouter, a network PC, a peer device or other common network node, andtypically includes many or all of the elements described above relativeto the computer 110, although only a memory storage device 181 has beenillustrated in FIG. 3. The logical connections depicted in FIG. 3include a local area network (LAN) 171 and a wide area network (WAN)173, but may also include other networks. Such networking environmentsare commonplace in offices, enterprise-wide computer networks, intranetsand the Internet.

When used in a LAN networking environment, the computer 110 is connectedto the LAN 171 through a network interface or adapter 170. When used ina WAN networking environment, the computer 110 typically includes amodem 172 or other means for establishing communications over the WAN173, such as the Internet. The modem 172, which may be internal orexternal, may be connected to the system bus 121 via the user inputinterface 160, or other appropriate mechanism. In a networkedenvironment, program modules depicted relative to the computer 110, orportions thereof, may be stored in the remote memory storage device. Byway of example, and not limitation, FIG. 3 illustrates remoteapplication programs 185 as residing on memory device 181. It will beappreciated that the network connections shown are exemplary and othermeans of establishing a communications link between the computers may beused.

One of ordinary skill in the art can appreciate that a computer 110 orother client devices can be deployed as part of a computer network. Inthis regard, the present invention pertains to any computer systemhaving any number of memory or storage units, and any number ofapplications and processes occurring across any number of storage unitsor volumes. An embodiment of the present invention may apply to anenvironment with server computers and client computers deployed in anetwork environment, having remote or local storage. The presentinvention may also apply to a standalone computing device, havingprogramming language functionality, interpretation and executioncapabilities.

Though the invention has been described in connection with certainpreferred embodiments depicted in the various figures, it should beunderstood that other similar embodiments may be used, and thatmodifications or additions may be made to the described embodiments forpracticing the invention without deviating therefrom. The invention,therefore, should not be limited to any single embodiment, but rathershould be construed in breadth and scope in accordance with thefollowing claims.

1. A logical gate for a quantum computer, the logical gate comprising:first and second electrically-conductive gates disposed within afractional quantum Hall effect (FQHE) fluid, the FQHE fluid having oneor more anti-dots disposed therein; and means for affecting a collectivestate of the one or more anti-dots by adjusting a voltage between thefirst and second gates to cause a quasi-particle to tunnel between thefirst and second gate.
 2. The logical gate of claim 1, wherein the FQHEfluid is a ν=5/2 quantum Hall effect fluid.
 3. The logical gate of claim1, wherein the quasi-particle is an anyon.
 4. The logical gate of claim1, wherein affecting the collective state of the one or more anti-dotscomprises converting the collective state from a state carrying trivialSU(2)—charge |1> to a state carrying a fermionic SU(2) charge |ε>. 5.The logical gate of claim 1, wherein affecting the collective state ofthe one or more anti-dots comprises converting the collective state froma state carrying a fermionic SU(2)—charge |ε> to a state carryingtrivial SU(2) charge |1>.
 6. A logical gate for a quantum computer, thelogical gate comprising: first and second electrically-conductive gatesdisposed within a fractional quantum Hall effect (FQHE) fluid, the FQHEfluid having one or more anti-dots disposed therein; and means foradjusting a voltage between the first and second gates to cause aquasi-particle to tunnel between the first and second gate.
 7. Thelogical gate of claim 6, wherein the FQHE fluid is a ν=5/2 quantum Halleffect fluid.
 8. The logical gate of claim 6, wherein the quasi-particleis an anyon.
 9. The logical gate of claim 6, wherein affecting thecollective state of the one or more anti-dots comprises converting thecollective state from a state carrying trivial SU(2)—charge |1> to astate carrying a fermionic SU(2) charge |ε>.
 10. The logical gate ofclaim 6, wherein affecting the collective state of the one or moreanti-dots comprises converting the collective state from a statecarrying a fermionic SU(2)—charge |ε> to a state carrying trivial SU(2)charge |1>.
 11. A gate reader for a quantum computer, the gate readercomprising: providing a fractional quantum Hall effect (FQHE) fluidhaving one or more anti-dots disposed therein; means for injecting aninput current into a first edge of a fractional quantum Hall effect(FQHE) fluid, the FQHE fluid having one or more anti-dots disposedtherein, such that the input current forms at least two tunnelingcurrent paths, the anti-dots being disposed between the tunnelingcurrent paths; means for removing from a second edge of the FQHE fluidan output current that represents a combination of the tunnelingcurrents; and means for determining from the output current a collectivestate of the one or more anti-dots.
 12. The gate reader of claim 11,wherein the means for determining the collective state of the one ormore anti-dots comprises means for determining whether the collectivestate of the one or more anti-dots corresponds to a state carryingtrivial SU(2)—charge |1>.
 13. The gate reader of claim 12, wherein themeans for determining the collective state of the one or more anti-dotscomprises means for determining whether the collective state of the oneor more anti-dots corresponds to a state carrying a fermionic SU(2)charge |ε>.
 14. The gate reader of claim 11, wherein the means fordetermining the collective state of the one or more anti-dots comprisesmeans for determining whether the collective state of the one or moreanti-dots corresponds to a state carrying a fermionic SU(2) charge |ε>.